This paper is structured as follows. We begin our discussion with the
recollection of known results about Stieltjes functions in
Section . In Section  we show that the
interpolation body \(V( z)\)  is shaped like a needle or maybe like a sword. (Our
language has an inadequate vocabulary limited to two and three-dimensional
shapes.) In Sections  and  we describe the algorithm.
The performance of the algorithm is demonstrated in Section  in
the context of electrochemistry, where the processes of corrosion and
electrolysis that occur in batteries and in many other natural and man-made
systems can be modeled by Voigt circuits—electrical circuits made only of
resistors and capacitors [1]}, [2]}, [3]}. The electrochemical
impedance spectrum (EIS) function \(Z(\omega )\)  can then be written as \(f(-i\omega )\) 
for some \(f\in \mathfrak {S}\) . Thus, the values \(Z_{j}=f(-i\omega _{j})\) ,
\(j=1,\ldots ,n\)  can be measured experimentally at particular frequencies
\(\omega _{1},\ldots ,\omega _{n}\) . Our algorithm takes noisy measurements of
\(Z_{1},\ldots ,Z_{n}\)  as the input and generates physically admissible EIS
function \(Z(\omega )\) , representing it both numerically and as the explicit
complex impedance function of a small Voigt circuit. It also displays the
certificate of optimality as well as the uncertainty of reconstruction of the
EIS function for the specific data. Figure REF  shows the typical
graphical output of the algorithm.
